Mathematical Sciences: Topological Properties of Singularities and Solutions of Nonlinear Equations

数学科学：奇点的拓扑性质和非线性方程的解

• 批准号: 9400930
• 负责人: James Damon
• 金额: $ 12.9万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400930&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topological; Properties; Singularities

9400930 Damon Professor Damon has been developing analogues of infinitesimal methods introduced by Mather for answering questions about the structure of nonlinear mappings and the topological structure of special but important classes of highly nonisolated singularities, such as occur in discriminants of mappings. In quite different circumstances, he has also discovered how to modify these infinitesimal methods for understanding the generic behavior of "pixel intensity functions" under Gaussian blurring used in computer vision. He intends to build on his earlier results and derive general algebraic formulae for certain fundamental topological invariants for highly singular spaces. Second, he will further develop methods for determining the generic properties of solutions to partial differential equations and, in particular, apply them to recent nonlinear versions of Gaussian blurring. Third, he will apply the results on the topology of singular spaces to understand how the qualitative behavior of the solutions can be deduced from the singularities which occur. This research project will investigate certain qualitative properties of the set of solutions of systems of nonlinear equations. Such properties will be applied to generally appearing solutions of partial differential equations. Professor Damon has been developing methods for determining such qualitative properties. Moreover, he has begun applying them to understand the effects of Gaussian blurring applied to "pixel intensity functions" for analyzing images in computer vision. He intends to develop these methods further so that they apply to general partial differential equations. This will include the investigation of relations between certain invariants of the solutions and their qualitative properties. One particularapplication will be to the increasingly sophisticated nonlinear blurring schemes which are being developed for identifying essential features and properties of images suc h as edges, ridges, etc. ***

9400930 Damon教授Damon一直在开发Mather引入的无限方法的类似物，以回答有关非线性映射结构的问题以及特殊但重要的高度非分离奇异性类别的拓扑结构，例如在映射的歧视者中发生。 在完全不同的情况下，他还发现了如何修改这些无穷小方法，以理解计算机视觉中使用的高斯模糊中“像素强度函数”的通用行为。 他打算以他的早期结果为基础，并为某些基本拓扑造型的高度奇异空间提供代数公式。 其次，他将进一步开发用于确定偏微分方程解决方案的通用属性的方法，尤其是将它们应用于最近的非线性版本的高斯模糊。 第三，他将对奇异空间拓扑的结果应用结果，以了解如何从发生的奇异性中推论解决方案的定性行为。 该研究项目将研究非线性方程系统解决方案解决方案的某些定性特性。 此类属性将应用于通常出现部分微分方程的解决方案。 达蒙教授一直在开发确定这种定性特性的方法。 此外，他已经开始应用它们来了解应用于“像素强度函数”的高斯模糊的影响，以分析计算机视觉中的图像。 他打算进一步开发这些方法，以便它们适用于一般的部分微分方程。 这将包括对解决方案的某些不变剂及其定性特性之间的关系的调查。 一个特殊的应用将是越来越复杂的非线性模糊方案，该方案正在开发出来，以识别图像的基本特征和特性作为边缘，山脊等。***